Theorem carden2b 8349

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8348 are meant to replace carden 8927 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	|- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 8347	. . . . 5 |- ( ( card ` B ) e. ( card ` A ) -> -. ( card ` B ) ~~ A )
2		ennum 8329	. . . . . . . 8 |- ( A ~~ B -> ( A e. dom card <-> B e. dom card ) )
3	2	biimpa 484	. . . . . . 7 |- ( ( A ~~ B /\ A e. dom card ) -> B e. dom card )
4		cardid2 8335	. . . . . . 7 |- ( B e. dom card -> ( card ` B ) ~~ B )
5	3, 4	syl 16	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ B )
6		ensym 7565	. . . . . . 7 |- ( A ~~ B -> B ~~ A )
7	6	adantr 465	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> B ~~ A )
8		entr 7568	. . . . . 6 |- ( ( ( card ` B ) ~~ B /\ B ~~ A ) -> ( card ` B ) ~~ A )
9	5, 7, 8	syl2anc 661	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ A )
10	1, 9	nsyl3 119	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` B ) e. ( card ` A ) )
11		cardon 8326	. . . . 5 |- ( card ` A ) e. On
12		cardon 8326	. . . . 5 |- ( card ` B ) e. On
13		ontri1 4912	. . . . 5 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
14	11, 12, 13	mp2an 672	. . . 4 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
15	10, 14	sylibr 212	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) C_ ( card ` B ) )
16		cardne 8347	. . . . 5 |- ( ( card ` A ) e. ( card ` B ) -> -. ( card ` A ) ~~ B )
17		cardid2 8335	. . . . . 6 |- ( A e. dom card -> ( card ` A ) ~~ A )
18		id 22	. . . . . 6 |- ( A ~~ B -> A ~~ B )
19		entr 7568	. . . . . 6 |- ( ( ( card ` A ) ~~ A /\ A ~~ B ) -> ( card ` A ) ~~ B )
20	17, 18, 19	syl2anr 478	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) ~~ B )
21	16, 20	nsyl3 119	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` A ) e. ( card ` B ) )
22		ontri1 4912	. . . . 5 |- ( ( ( card ` B ) e. On /\ ( card ` A ) e. On ) -> ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) ) )
23	12, 11, 22	mp2an 672	. . . 4 |- ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) )
24	21, 23	sylibr 212	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) C_ ( card ` A ) )
25	15, 24	eqssd 3521	. 2 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) = ( card ` B ) )
26		ndmfv 5890	. . . 4 |- ( -. A e. dom card -> ( card ` A ) = (/) )
27	26	adantl 466	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = (/) )
28	2	notbid 294	. . . . 5 |- ( A ~~ B -> ( -. A e. dom card <-> -. B e. dom card ) )
29	28	biimpa 484	. . . 4 |- ( ( A ~~ B /\ -. A e. dom card ) -> -. B e. dom card )
30		ndmfv 5890	. . . 4 |- ( -. B e. dom card -> ( card ` B ) = (/) )
31	29, 30	syl 16	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` B ) = (/) )
32	27, 31	eqtr4d 2511	. 2 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = ( card ` B ) )
33	25, 32	pm2.61dan 789	1 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3476   (/)c0 3785   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5588   ~~ cen 7514   cardccrd 8317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7312  df-en 7518  df-card 8321

This theorem is referenced by:  card1  8350  carddom2  8359  cardennn  8365  cardsucinf  8366  pm54.43lem  8381  nnacda  8582  ficardun  8583  ackbij1lem5  8605  ackbij1lem8  8608  ackbij1lem9  8609  ackbij2lem2  8621  carden  8927  r1tskina  9161  cardfz  12049